View source for Dual functions

Functions complementary in the sense of Young, i.e. strictly convex functions (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]) connected by the [[Legendre transform|Legendre transform]].

====Comments====
For certain real-valued non-decreasing functions defined on the positive half-line (including zero) there is a natural notion of an inverse. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d0341001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d0341002.png" /> are such inverses to each other, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d0341003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d0341004.png" /> defined (on the positive half-line) by

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are said to be complementary in the sense of Young or Young-conjugate. For them Young's inequality holds:

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Associated with a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d0341007.png" /> of non-vanishing functions complementary in the sense of Young and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d0341009.png" />-finite measure, there is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410011.png" /> of complete normed spaces. These spaces, consisting of (equivalence classes of) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410012.png" />-measurable functions, are called Orlicz spaces (cf. [[Orlicz space|Orlicz space]]). The Lebesgue spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410013.png" /> (cf. [[Lebesgue space|Lebesgue space]]) are particular cases of Orlicz spaces, cf. [[#References|[a4]]].

In a more abstract setting, the name dual functions is reminiscent of [[Dual pair|dual pair]] in [[Duality|duality]] theory and of dual problems in [[Convex programming|convex programming]] and optimal control (cf. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), but this name is rarely used in English: the most common name is (convex) conjugate functions (cf. [[Conjugate function|Conjugate function]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410015.png" /> be two real vector spaces in separate duality with respect to a bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410016.png" /> (the usual one if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410017.png" />), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410018.png" /> be a mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410019.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410020.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410021.png" /> is only defined on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410023.png" />, set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410025.png" />, the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410026.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410027.png" /> is non-empty, the dual, or polar, or adjoint, or better conjugate function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410028.png" /> is the convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410029.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410030.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410031.png" /></td> </tr></table>

The following result is a generalization of the geometric [[Hahn–Banach theorem|Hahn–Banach theorem]] on the bipolar of a set: the biconjugate function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410033.png" /> is the greatest lower semi-continuous convex function bounded above by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410034.png" />, and so is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410035.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410036.png" /> is a lower semi-continuous convex function (in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410037.png" /> is called a pair of conjugate functions). The notion of conjugate function, which was introduced by W. Young in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410038.png" /> and by W. Fenchel in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410039.png" />, is very important in convex analysis; it is closely related to the notion of [[Subdifferential|subdifferential]]: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410040.png" /> is convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410041.png" /> is its subdifferential, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410043.png" /> one has

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This can be written, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410045.png" /> is lower semi-continuous, as

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The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410047.png" /> is often called the Fenchel transform, sometimes with the name of Young or Legendre, or both, added. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410049.png" /> is sufficiently smooth as a convex function, it is a special case of the [[Legendre transform|Legendre transform]]; on the other hand, it is also a special case of a [[Galois correspondence|Galois correspondence]]; these facts are of secondary importance in convex analysis. The notion of conjugate function plays a fundamental role in convex optimization. It is used to define the Lagrangian of some problem and the associated dual problem.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410050.png" />, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410051.png" /> is called a Young function if it is a non-decreasing convex function such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410053.png" />. The conjugate function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410054.png" /> of a Young function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410055.png" /> is still a Young function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410056.png" />; for example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410057.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410059.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410060.png" /> is the conjugate exponent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410061.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410062.png" />. Young functions are used to define Orlicz spaces (cf. [[Orlicz space|Orlicz space]]), and pairs of conjugate Young functions are used to study the duality between them; more generally they help to establish various inequalities in measure theory (Burkholder inequalities in martingale theory, Chernov's inequality in classical probability theory, Kullback's inequality in statistics, etc.), via the easy to prove but fundamental Young inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034100/d03410063.png" /></td> </tr></table>

which enabled Young to solve a problem about Fourier transformation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.T. Rockafellar,   "Conjugate duality and optimization" , ''Reg. Conf. Ser. Appl. Math.'' , SIAM  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neveu,   "Martingales à temps discret" , Masson  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer,   "Probabilities and potential" , '''2. Theory of martingales''' , North-Holland  (1978–1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.C. Zaanen,   "Linear analysis" , North-Holland  (1956)</TD></TR></table>